Abstract
The bending and free vibrational behaviors of functionally graded (FG) cylindrical beams with radially and axially varying material inhomogeneities are investigated. Based on a high-order cylindrical beam model, where the shear deformation and rotary inertia are both considered, the two coupled governing differential motion equations for the deflection and rotation are established. The analytical bending solutions for various boundary conditions are derived. In the vibrational analysis of FG cylindrical beams, the two governing equations are firstly changed to a single equation by means of an auxiliary function, and then the vibration mode is expanded into shifted Chebyshev polynomials. Numerical examples are given to investigate the effects of the material gradient indices on the deflections, the stress distributions, and the eigenfrequencies of the cylindrical beams, respectively. By comparing the obtained numerical results with those obtained by the three-dimensional (3D) elasticity theory and the Timoshenko beam theory, the effectiveness of the present approach is verified.
Highlights
Compared with traditional laminated materials, functionally graded (FG) materials can significantly reduce the risk for delamination failures
If a Timoshenko or higher-order beam model is adopted to investigate the free vibrations of axially FG beams, two coupled governing differential equations with variable coefficients will be derived, which makes it difficult to obtain the exact solutions due to the arbitrary gradient changes
Based on the differential transformation element method, Rajasekaran and Tochaei[13] studied the free vibrations of axially FG Timoshenko beams
Summary
Compared with traditional laminated materials, functionally graded (FG) materials can significantly reduce the risk for delamination failures. Based on the finite element method, Shahba et al.[11] discussed the free vibrations of arbitrarily tapered and axially FG Timoshenko beams under various boundary conditions. Huang et al.[12] introduced a unified method to analyze the free vibrations of Timoshenko beams with arbitrarily axial material parameters by transforming the governing equations into a system of linear algebraic equations under different boundary conditions. Based on the differential transformation element method, Rajasekaran and Tochaei[13] studied the free vibrations of axially FG Timoshenko beams. Tang et al.[14] studied the free vibrations of non-uniform FG Timoshenko beams, and derived the frequency equations in closed form, where the material properties were assumed to vary in a unified exponential law. Zhang et al.[16] presented an effective approximation to investigate the free vibrations of axially FG beams based on the Euler beam theory and the Timoshenko beam theory, respectively
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